H-bridges at that current level are available, just a little pricey and possible larger then will fit in your car. If you don't need reverse for the car then speed control can be simple and inexpensive using a logic level mosfet transistor of suitable voltage and current rating and driving it with a arduino pwm output pin. It's not known to us how your car presently controls it's motor speed and if it could be modified for Arduino control or not.

10A+ DPDT relays will reverse direction (occasionally and not quickly; probably fine for fwd/reverse.) It's probably a toss-up whether that would work out cheaper than a "real" H-bridge...

You could control forward and reverse that way, but without adding an extra relay, you couldn't control on/off! So, it would be better to use a couple of SPDT relays instead of a single DPDT...

You wouldn't have PWM control, but you might be able to put on the low side a power FET or transistor, and PWM that, and switch direction with the relays; the whole thing could probably be done under $15.00 USD...

I will not respond to Arduino help PM's from random forum users; if you have such a question, start a new topic thread.